I am "struggling" with a basic question, and would like someone to help me to point out what's wrong in my logic.
Let A is a subset of vectors in V(4,3): A = {(0,1,2,1), (1,0,2,2), (1,1,1,0), (1,2,0,1)}
- Is A linearly independent? Find the 2 bases for < A >.
My approach is that, consider: a(0,1,2,1) + b(1,0,2,2) + c(1,1,1,0) + d(1,2,0,1) = (0,0,0,0) Then we have a system of 4 equations:
b + c + d = 0
a + c + 2d = 0
2a + 2b + c = 0
a + 2b + d = 0
Solve for a, b, c and d. Then we got a = b = c = d = 0, Hence A is linearly independent.
However, the solution given by my math professor is that:
Since 1(0, 1, 2, 1) + 1(1, 0, 2, 2) + 2(1, 1, 1, 0) + 0(1, 2, 0, 1) = (0, 0, 0, 0), X is linearly dependent
To me, clearly it's not, as: 1 + 2 #0, 1 + 2 # 0, 2 + 2 + 2 # 0 and 1 + 2 # 0.
Can someone help me to understand why? And also, what is an easy way to determine if a set of vectors is linearly independent or not? As, my approach, solving a system equations can be very time consuming, so not very practical in the test or exam environment.
Many thanks.